This study uses the LES data of two shallow cumulus cloud fields forming the basis for the synthetic satellite retrievals. The chosen LES scenes originate from the study of , and are described in detail therein. Here, we only recap the essential settings relevant for this study and elaborate on the simplifications that have been made to the input.

The LES has been performed using MicroHH . The full model setup for the current study can be found in . In MicroHH radiation is solved in 1D using RTE-RRTMPG (Radiative Transfer for Energetics + RRTM for General circulations model applications–Parallel; ) or in 3D using the raytracer of (see Sect. ). The initial and boundary conditions are based on ERA5 and have been derived using LS2D . The horizontal size of the domain is 25.6 × 25.6 km with a resolution of 50 m. Vertically, the model extends up to 6.4 km with a resolution of 25 m. MicroHH accounts for the impact on radiation of gases located above the domain top by computing radiation in a single column that extends up to the top of the atmosphere.

The first of the two selected scenes is taken on 25 March 2014 at 12:00 UTC. The scene is characterised by a broken cumulus cloud field over grasslands, which is representative of the area around the Cabauw measurement tower in the Netherlands (51.4° N 4.9° E). The cloud cover of the scene is 22 %. The cloud base and cloud top heights, as determined from the lowest and highest level with a non-zero liquid water content, vary over the domain. The domain-averaged cloud base and top heights are 1550 and 1867 m, respectively. At 12:00 UTC the Sun is approximately south (the solar azimuth angle is 184.2°, measured clockwise from the North) with a solar zenith angle of 50.53°. In the remainder of the article, these values will be presented rounded to the nearest integer.

The second LES scene is taken on 4 May 2016 at 14:00 UTC, for the same domain as scene 1. Compared to the first scene, there is a higher cloud cover of 64 %. Compared to scene 1, the domain-averaged cloud base and cloud top are slightly higher, at 1884 and 2126 m, respectively. For a better comparison between scenes, we chose to stick to the solar geometry of scene 1, and also use the same background profiles of temperature, pressure, and ozone. Hence, the only model variables adjusted for scene 2 are the cloud properties re and liquid water path (LWP) and the water vapour content. Both scenes consists fully of water phase cloud droplets.

To make the LES data consistent with the assumptions made in the code to simulate TOA reflectances (i.e. MONKI, see Sect. ) and the satellite retrieval (i.e. CPP-SICCS, see Sect. ), some simplifications have been made to the original LES scenes.

First, in the original, realistic, LES scenes of aerosols are included. Because we want to limit the study to 3D cloud effects only, all aerosols are excluded from the simulation. Concerning gases in the atmosphere, in MONKI, we only assume ozone as an absorbing gas, relevant at 0.640 µm, the visible (VIS) wavelength of the simulation. In the satellite retrieval, the effects of water vapour and CO₂ on GHI are included in addition to ozone. All other gases in the LES data are ignored.

To be consistent with the satellite retrieval, mid-latitude summer atmospheric profiles from are assumed for ozone, pressure, and temperature instead of the original profiles from the LES scenes. Furthermore, the cloud droplet effective radii of the scene have been rounded to their nearest integer values, in order to align them to the droplet sizes used in MONKI (see Sect. ). Finally, based on LES data a cloud mask is defined by LWP > 0 kg m⁻².

To create synthetic data satellite retrievals of TOA reflectance, the accurate and thoroughly validated 3D radiative transfer code Monte Carlo KNMI MONKI; is used. MONKI can simulate Rayleigh scattering by gases, Mie scattering by clouds, gaseous and cloud absorption and Lambertian surface reflection, fully taking into account polarisation of light for all orders of scattering. For Earth-based applications, the optical properties of MONKI are identical to Doubling Adding KNMI (DAK), the RT code that is used by the satellite retrieval see Sect. . To enable retrieval of cloud properties and GHI, TOA reflectances are simulated with MONKI at wavelengths of 0.640 and 1.625 µm. These wavelengths correspond to respective SEVIRI VIS/near-infrared (NIR) channels and allow for bispectral retrieval of cloud properties . In contrast to the actual reflectances from SEVIRI, which cover a narrow spectral band, the MONKI reflectances are fully monochromatic.

The MONKI simulations are performed using 10⁵ photons per column. In the simulations, the 3D state of the atmosphere is considered, and in addition to total radiative intensity, linear polarization is taken into account. MONKI simulations require atmospheric profiles for temperature, pressure, and ozone volume mixing ratios, for which mid-latitude summer profiles are assumed . Furthermore, MONKI requires the cloud optical depth (τ) and effective cloud droplet radius (re) in µm for each cloudy grid cell. τ has been derived from the LES data using the liquid water path (LWP) in kg m⁻² and re in m, following : 1τ=34QextLWPρliqre. Here, the liquid water density (ρliq) is 10³ kg m⁻³, and Qext represents the wavelength-dependent extinction efficiency, a unitless quantity derived from Mie theory, with a value close to 2. The range of available re in the current MONKI simulations is 3 to 22 µm in steps of 1 µm. This step size was used since including droplet sizes with smaller steps would make the MONKI simulations computationally unfeasible. However, for the current scenes all re are between 3 and 11 µm. Finally, per re, MONKI requires information on scattering properties, which has been generated with version 3.1 of the Meerhof Mie Program , using a two-parameter gamma droplet size distribution with an effective variance of 0.10.

In addition to the solar angles prescribed in the LES data file, MONKI requires information on satellite viewing and zenith angle. In the current simulations, a satellite viewing angle of 0° is used. For geostationary satellites observing the mid-latitudes, a nadir satellite viewing angle is not realistic. However, at slanted viewing angles, retrievals will be increasingly affected by parallax. The reference simulations are independent of viewing angle and therefore not influenced by parallax. Using a viewing zenith angle of 0° ensures that the retrieval does not require corrections for parallax, which would cause additional uncertainties. Note that for polar orbiting satellites such as MODIS, nadir viewing angles can be valid for the selected latitudes.

To retrieve GHI from synthetic MONKI simulated TOA reflectances, a two-step approach is followed. First, τ and re are retrieved using the Cloud Physical Properties (CPP) algorithm , which matches the synthetic VIS and NIR reflectances to simulated reflectances of homogeneous clouds, generated with DAK and stored in look-up tables (LUTs). Since the cloud field fully consists of water-phase clouds, only water clouds are considered in the retrieval. As mentioned in Sect. , the MONKI simulations are fully monochromatic, including only absorption by ozone, and the cloud property retrievals are performed correspondingly. Pixels are flagged as cloudy if the retrieved τ is above zero. Note that this internal retrieval cloud mask differs from the LES based cloud mask.

The second step is the retrieval of GHI using the Solar Irradiance under Clear and Cloudy Skies (SICCS) algorithm . SICCS is based on radiative transfer simulations with a broadband version of DAK covering the spectral range 0.240 to 4.606 µm and using the correlated k-distribution technique to account for atmospheric gas absorptions. Cloudy-sky GHI is calculated for all cloudy pixels, and clear-sky GHI is calculated for all pixels.

In line with the MONKI simulations, no aerosols are included in these retrievals. Note that the atmosphere in DAK is plane-parallel, so the retrieved GHI with CPP-SICCS does not include 3D cloud effects.

As input, CPP-SICCS requires the cosine of the solar and satellite zenith angles, the relative azimuth between the Sun and the satellite, the surface albedo for both the VIS and NIR channels, the water vapour column per pixel in kg m⁻², and the ozone concentration in Dobson units (DU). In the retrieval the ozone concentration is fixed at 332 DU, which is a typical value for the mid-latitudes in summer and the same as used in the MONKI simulations.

In addition to GHI retrieved from the simulated TOA MONKI reflectances and the CPP-SICCS algorithm, GHI is also computed directly from the LES data using a Monte Carlo Raytracer hereafter, MCRT;. MCRT is an extension of the C++/CUDA implementation of the 1D RT model RTE-RRTMPG. In contrast to the CPP-SICCS satellite retrieval, where the atmosphere is plane-parallel and the different columns are independent, MCRT enables the simulation of both 1D, that is, assuming plane-parallel and independent pixels, as well as full 3D RT. The version of MCRT employed in this study uses another set of Mie look-up tables to determine the scattering directions for water clouds with an re between 3–22 µm in steps of 1 µm. To compute broadband fluxes, the raytracing is performed independently over 14 wavelength bands, each containing 224 quadrature points. The total spectral width covered by MCRT ranges from 0.2 to 12.2 µm. which is broader than for the satellite retrieval. However, since the spectral differences occur at the edges of the solar spectrum where a negligible amount of solar radiation is received, this results in negligible differences in broadband GHI. In this study, MCRT is run using 1024 samples per pixel per quadrature point.

In addition to the simulations of the two default scenes introduced in Sect. , five additional sensitivity experiments are performed for the scene with low cloud cover (Scene 1). With the first sensitivity test, the importance of changes in surface albedo is studied by using a surface albedo of zero (black surface) instead of the default value of 0.22. In the second sensitivity experiment the influence of solar zenith angle (θ0) variations is studied by reducing θ0 from 51° to 15°. The third experiment combines adjustments of the previous two sensitivity tests. The last two sensitivity experiments are aimed at the influence of the viewing zenith angle (θ) on retrieval accuracy. In this experiment the satellite position is changed to a satellite viewing angle of 40° or 70° with a relative azimuth of 60° between the Sun and the satellite. An overview of the used geometry and surface albedo for various simulations is presented in Table . Hereafter, we will refer to the simulations with different geometries or surface albedo as scenarios.

To study the influence of spatial resolution and 3D cloud effects on GHI retrieval accuracy, we retrieve and simulate GHI using four main pathways, shown graphically in the flow chart of Fig. . In the retrieval pathways, MONKI is used to simulate TOA reflectances that serve as input for the CPP-SICCS GHI retrieval (Top pathways of Fig. ). In the reference pathways, MCRT is used to directly compute GHI from the LES scene (bottom pathways of Fig. ). To be able to isolate 3D cloud effects, both MCRT and MONKI are run using 3D RT and 1D RT, i.e., assuming independent pixels. This results in four GHI fields per scenario as summarised in Table . The four pathways are also conceptually illustrated in Fig. .

Next, to study the effects of spatial resolution on retrieval accuracy, the 0.05 km pixel sizes of the original LES scene are spatially averaged by a factor 2 using box-averaging, as shown by the square boxes in Fig. . Spatial averaging is repeated multiple times to assess retrieval accuracy at 9 different spatial resolutions, namely: 0.05, 0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, and 12.8 km. Note that for the reference pathway the GHI output is spatially averaged. However, for the retrieval pathway we do not average GHI but rather the MONKI TOA reflectances and perform the satellite retrieval based on the averaged reflectances. For the retrieval, averaging the reflectances is representative of what a real instrument would measure at the selected spatial resolution.

At each resolution, the errors between the retrieval and the reference can be separated into various components. First, the error resulting from the IPA can be derived from the comparison of the 1D reference GHI with the 3D reference GHI, as indicated by the vertical rectangles in the bottom right of Fig. . The IPA error is caused by the neglect of 3D RT.

Next, the PPA error can be quantified by comparing the 1D retrieval to the 1D reference (shown by the vertical rectangles in the middle right of Fig. ). The PPA error is introduced by spatial averaging of reflectances in the satellite retrieval. Since both 1D pathways are fully plane-parallel within each pixel and the IPA is used, 3D radiative effects are not present, and thus the error must originate from ignoring the sub-pixel variability. At the original spatial resolution of 0.05 km, no spatial averaging is performed, and therefore the corresponding PPA error should be 0. This is only valid in case the MCRT reference simulation and the MONKI + CPP-SICCS satellite retrieval are consistent. In Sect. the consistency of the models is validated. Note that potential sub-pixel variability at spatial scales finer than 0.05 km cannot be captured by our simulations but does occur within real cloud fields. We neglect potential sub-pixel cloud variability at spatial scales finer than 0.05 km.

Finally, we identify a residual error from the comparison between the 3D retrieval and the 1D retrieval. In Fig. this is illustrated by the vertical rectangles in the upper right corner. This residual error can be explained as follows. In the 1D retrieval both the simulation of reflectances and retrieval of GHI are performed relying on the IPA, which makes these steps fully consistent. For the 3D retrieval, independent pixels are assumed in the GHI retrieval step but not for the simulation of TOA reflectance, which is fully 3D. As a result, inconsistencies between the simulation of reflectances (with MONKI) and retrieval of GHI (with CPP-SICCS) are introduced.

In the current study, we quantify errors using the bias and root mean square error (RMSE) as evaluation metrics. Equation shows for the bias how the three components of the error can be derived from the pathways noted in Table and shown in Fig. and how they add up to the total bias, 2(GHIRetrieval3D-GHIReference3D)︸Totalbias=(GHIReference1D-GHIReference3D)︸IPA bias+(GHIRetrieval1D-GHIReference1D)︸PPA bias+(GHIRetrieval3D-GHIRetrieval1D)︸Residual bias.

To rule out inconsistencies between the forward radiative transfer models (MONKI and MCRT) and the retrieval codes (CPP-SICCS), a validation is performed for cloud properties and GHI based on the 1D retrieval and 1D reference at a spatial resolution of 0.05 km.

The results presented so far were for a cumulus scene with a relatively low cloud cover of 22 % (scene 1). To study the influence of cloud cover on the retrieval errors, this section presents the analysis for a cumulus cloud field with a higher cloud cover of 63.9 % (scene 2). The spatial distribution of the 3D retrieved and 3D reference GHI at the finest spatial resolution of 0.05 km is presented in Fig. . These spatial distributions show that compared to scene 1, this scene is characterised by larger cumulus clouds also leading to fewer but larger cloud shadows. In the end, at a 0.05 km spatial resolution, the cloud shadow fraction between scene 1 and scene 2 are comparable (compare Fig. d to Fig. c). Furthermore, visual inspection of Fig. reveals that a smaller fraction of irradiance enhanced pixels is retrieved. Despite the smaller fraction of irradiance enhanced pixels, the absolute magnitude of the enhancement is larger compared to scene 1, which is illustrated in the histograms of Fig. a.

Overall, the distribution of 3D retrieved GHI for scene 2 follows a similar distribution as the 3D retrieved GHI from scene 1 (compare solid lines of Figs. e and a). The main difference from scene 1 is that because of the higher cloud cover, the peak in the PDF corresponding to the retrieval error due to irradiance enhancement remains more limited. Moreover, the peak occurs at slightly lower normalised GHI values. This is indicative of stronger irradiance enhancement occurring in the high cloud cover scene leading to correspondingly larger reductions in GHI. The stronger magnitude of the irradiance enhancement for scene 2 is confirmed by the 3D reference, with enhancement going up to 35 % to 40 % (compare dashed lines of Figs. e and a). Similarly to scene 1, for the 3D reference GHI a bimodal PDF is observed, but due to the larger fraction of cloudy and shaded pixels, the diffuse peak of scene 2 has a higher probability density.

In Fig. b, the domain averaged normalised GHI is illustrated. Generally, the trends are comparable to the corresponding low cloud cover scenario (compare Fig. b to Fig. d). Due to the higher cloud cover the absolute magnitude of domain averaged GHI is lower than for scene 1. For the 1D and 3D reference simulations, the domain averaged GHI is again preserved moving towards coarser spatial resolutions.

Next, as a result of the clear-cloudy mixing, the domain averaged GHI of the 1D retrieval reduces when the spatial resolution gets coarser (triangular-markers, Fig. b). Finally, just as for the scenario with low cloud cover, the 3D retrieved GHI increases towards a resolution of 1.6 km, after which it starts to decrease again (cross-markers, Fig. b). The difference from scenario 1 is that here the increase in GHI remains more limited, whereas the decrease in domain averaged GHI at coarser spatial resolutions is more substantial.

This can be explained again by considering the balance between the errors due to irradiance enhancement, shading, and the clear-cloudy mixing. As the initial fraction of irradiance enhanced pixels at 0.05 km is considerably lower than for scene 1 (Fig. c), the error due to irradiance enhancement is lower accordingly and GHI is closer to the reference. With decreasing resolution, the fraction of irradiance enhanced pixels decreases resulting in increasing domain averaged GHI. However, note that although the fraction of irradiance enhanced pixels is lower, the magnitude of the enhancements is actually larger for the high cloud cover scene (compare the solid lines from Fig. e with Fig. a). The result is a slower cancellation of the cloud enhanced pixels at decreasing spatial resolutions (Fig. c). Consequently, the increase in domain averaged GHI remains more limited toward coarser resolution. At a spatial resolution of 1.6 km there is a sharp decrease in domain averaged GHI. With both the irradiance enhanced and shaded pixels disappearing, clear-cloudy mixing becomes the most important error, and therefore, the domain averaged GHI is decreasing again.

From the GHI bias plots we can observe that the trends for the IPA and PPA related bias and residual bias (Fig. a) are comparable to those of the corresponding scenario of scene 1 (See Fig. d). Furthermore, also the total bias is in line with the total biases as observed for scene 1, with the smallest biases at a spatial resolution of 1.6 km. Contrary to scene 1, here, the biases at coarser spatial resolutions are larger than at the finest resolutions.

In terms of RMSE (Fig. b), comparing scene 1 and scene 2 shows smaller PPA and residual errors for scene 2. In contrast, for the IPA at 0.05 km spatial resolution the error is about 65 W m⁻² larger for scene 2 than for scene 1. Consequently, the IPA error remains dominant until coarser spatial resolutions. Only around a spatial resolution of 5 to 6 km does the PPA error become larger than the IPA error, while this occurs at 2 to 3 km spatial resolution for scene 1.

To illustrate how clear-cloudy mixing can introduce transmittance biases in the retrieval via reflectance averaging, we assume a new, highly idealised cloud field consisting of just two pixels in this section. The first pixel has a very low optical depth of 0.25, and the other pixel has a τ of either 5, 25 or 90. The satellite and solar geometry of this idealised scenario are taken directly from the DAK LUT (see Sect. ) and closely match the default used in this study (Table ).

In Fig. , for the chosen geometry, the relation between reflectance (solid blue curves) and optical depth as well as transmittance (dash dotted green curves) and optical depth are presented for the three optical depth contrasts between τ=0.25 and τ=5 (Fig. a and d), τ=25 (Fig. b and e) or τ=90 (Fig. c and f), respectively. In Fig. 15a, b, and c the narrowband TOA reflectance at 0.64 µm is plotted, while in Fig. d, e, and f the (hemispheric) broadband TOA albedo is shown.

The relation between τ and reflectance is highly nonlinear, as demonstrated by the solid, blue curves in Figure . As a result, coarsening of the the spatial resolution by averaging the retrieved reflectances of the two pixels (indicated by the open blue circles on the reflectance curves), leads to a substantially lower τ than would have been obtained if, instead of reflectances, the optical depth of both pixels would have been averaged (indicated by the crosses on the dashed lines between the reflectance curves).

With the optical depth determined, the next step is to derive surface irradiance. For this, the relation between τ and transmittance is used (indicated by the green dashed-dotted curves). This relation is also highly nonlinear, causing another inhomogeneity error. However, when the narrowband TOA albedo is used to retrieve τ, the relation between transmittance and optical depth is almost exactly the inverse of that between τ and TOA albedo (Figs. d, e, f). Hence, if broadband TOA albedo observations are used to retrieve τ and subsequently transmittance, the respective errors cancel out see also: the retrieved transmittance for the combined pixel (indicated by the open green circles on the transmittance curves) is very close to the average of the retrieved transmittance for the individual pixels (indicated by the green crosses on the the dashed lines between the transmittance curves).

However, the current study does not use broadband (hemispheric) TOA albedo to derive τ but rather narrowband TOA reflectances, corresponding to measurements from satellite imagers. For these non-hemispheric observations, the relation with τ (solid lines in Figs. a, b, c) is slightly different from that for the broadband TOA albedo (solid lines in Figs. d, e, f). As a result, depending on the geometry, the τ retrieval offset is not fully compensated for by the transmittance retrieval. Indeed, the retrieved transmittance at coarse resolution (open green circles in Fig. a, b, c) is lower than the actual value (green crosses). Notably, if the clear-cloudy contrast grows, the magnitude of this negative transmittance bias grows accordingly, as indicated by the larger deviation between the horizontal, yellow, dotted lines in the middle and right columns.

Since the hemispheric albedo leads to unbiased retrievals of transmittance (Fig. d, e, f), there must be specific Sun-satellite geometries for which positive transmittance biases occur as a result of the clear-cloudy mixing. In Fig. , the transmittance bias is shown as a function of θ and Δϕ for two values of θ0 and four τ-contrasts. From this figure the following observations can be made. Firstly, at small satellite viewing angles, negative transmittance biases are observed, which are compensated by positive biases at high θ (above ≈ 60°). Secondly, as θ0 grows from 20 to 50°, the overall biases, either negative or positive, increase as well, and positive transmittance biases start to occur already at lower θ. Next, as is also shown in Fig. , if the clear-cloudy contrast gets larger, this will generally lead to stronger transmittance biases. Moreover, the rightmost column of Fig. demonstrates that clear-cloudy mixing does not lead to major transmittance biases for moderately thick to thick clouds (τ ≳ 5). In other words: the clear-cloudy mixing errors are related to retrieval at low τ. Figure does not show the effect of re on the transmittance bias but this effect is marginal.

With the relation between clear-cloudy mixing and viewing geometry pointed out in Sect. , here the scenarios are presented for viewing angles of 40 and 70°. By evaluating Fig. for these two scenarios (also see Table ), either limited (θ = 40°) or positive (θ = 70°), clear-cloudy mixing biases are expected.

First, for both slanted viewing angles, the PDF of the 1D retrieval is again dominated by high probability densities for the clear-sky pixels (dotted lines, Fig. a, d). For the retrieval based on 3D reflectances, clearer deviations are apparent compared to the default scenario. The peak in probability density corresponding to the clear-sky pixels that are incorrectly retrieved as cloudy is closer to the actual clear-sky value than for the default scenario (Compare solid lines of Fig. a, d to Fig. e). This indicates a smaller error due to irradiance enhancement as the viewing angle increases. Although the peak itself is narrower, in the scenario where θ = 70°, the peak's base is wider than in the default scenario, which we attribute to parallax.

Parallax is the shift between the actual position (from the LES data) and apparent (retrieved) position of an observed object (i.e. clouds). Until this section, all retrievals were performed for nadir viewing angles for which there is no parallax. Slanted viewing results in a shift of the projected location of the clouds in the direction away from the observer. The magnitude of parallax depends on the height of the cloud and the viewing angle . Larger displacements occur for higher clouds or at larger viewing zenith angles. For the retrievals with slanted viewing angles, each pixel will have variable parallax due to variations in cloud occurrence and vertical structure. As a result, clouds that are projected onto a single pixel at nadir viewing might be projected onto multiple pixels at a higher θ, leading to smoother reflectances. If reflectances are smoothed, the retrieved cloud edges become less sharp, resulting in a smoother transition from clear to cloudy in the PDF for retrieved GHI. Due to smoothing, the retrieved cloud fraction effectively increases (not shown). Another effect of the smoothing is that both the fraction of irradiance-enhanced and cloud-shaded pixels is reduced, also reducing the retrieval error due to shading and irradiance enhancement. On the other hand, parallax does introduce an additional mismatch, as clear-sky pixels will be incorrectly classified as cloudy by the retrieval.

At a θ of 40° the change in domain-averaged GHI remains limited (triangular markers in Fig. b). Meanwhile, for θ = 70°, an increase in domain-averaged GHI is observed for the 1D retrieval as the resolution gets coarser (triangular markers in Fig. e). Both observations are as expected because with a θ of 40°, the clear-cloudy mixing bias remains limited and for θ = 70°, even a positive GHI bias is expected due to clear-cloudy mixing (see Fig. ).

For the retrievals based on 3D reflectances (cross markers in Fig. b, e), towards coarser resolutions, there is an increase in domain-averaged GHI as a result of the increase in relative importance of the retrieval error due to shading over the retrieval error due to irradiance enhancement. However, at coarser spatial resolutions, for both viewing angles, no decrease in domain-averaged GHI is observed as was the case for the default scenario of scene 1. At these coarser resolutions, the influence of irradiance enhancement and shading are limited. Moreover, for these geometries, at coarser resolutions, changes due to clear-cloudy mixing are limited as well (see 1D retrieval Fig. b, e), and therefore GHI remains unchanged towards even coarser resolutions.

Finally, for the biases (Fig. c, f), the IPA-related biases are identical to the default scenario. Since the averaged GHI of the references is constant with resolution, the PPA-related bias follows the same trend as the 1D retrieval. Next, both at a θ of 40° and a θ of 70° a positive trend in residual bias is observed if spatial resolution gets coarser. However, contrary to the scenarios with a nadir view and varying solar zenith angles, albedo, and cloud cover, for the scenario with a θ of 70°, the residual bias remains negative at all spatial resolutions. Overall, for this viewing angle, this results in a negative total bias which vanishes for spatial resolutions of 1.6 km and coarser. Interestingly, at a θ of 40° the total bias is minimal, already at a spatial resolution of 0.2 km. For most other scenarios the total bias reaches its minimum at spatial resolution between 1 and 3 km. What the scenario of θ = 40° indicates is that the observed minimal biases around 1 km, are not a general rule but heavily depend on the present geometry.

The current study remains limited to two variable shallow cumulus cloud fields. Results do not necessarily generalise to other cloud conditions. Likely, in more homogeneous scenes, both IPA and PPA errors will be smaller. Future studies should clarify the influence of 3D cloud-radiation interactions on retrieval accuracy in those conditions.

When generalising the presented results, not only cloud conditions should be considered. As we show in the current study, surface albedo and viewing geometry have a considerable influence on the spatial resolution at which the total bias is minimal. While the minimal total bias of the default scenario of scene 1 occurs at a spatial resolution of 1.6 km, this reduces to 0.4 (surface albedo = 0.0) or even 0.2 km (θ = 40°) in two of the sensitivity experiments.

Furthermore, in terms of RMSE, at resolutions finer than 2 to 6 km, IPA errors are the most important contributor to the overall error. Also, this balance might shift for other cloud conditions or viewing geometries. For instance, in situations with higher cloud cover and a large optical depth variability, relatively larger PPA contributions might be expected.

Although the different error components may vary strongly across cloud conditions and viewing geometries, the mechanisms describing these errors, as explained in this article, remain valid under these varying conditions, albeit with a different partitioning. Here, we focused on a single retrieval algorithm (i.e., CPP-SICCS), however, similar results would likely have been found using other 1D physics-based retrieval algorithms.

The scenarios presented in this study have been highly idealised, enabling the illustration of the first-order effect of retrieval errors due to spatial inhomogeneity and 3D radiative transfer effects. Compared to non-synthetic satellite retrievals, some deviations are expected.

For instance, for the current simulations, aerosols were excluded. Although the overall radiative effects of aerosols are much smaller than those of clouds, they still influence GHI. In the presence of aerosols, there will be a stronger extinction of irradiance from the direct beam, especially near cloud edges due to aerosol hygroscopic growth, also leading to increased scattering of radiation into cloud shadows . For satellite retrievals, increased scattering would result in stronger irradiance enhancement errors in clear-sky regions. On the other hand, increased scattering towards cloud-shaded regions would reduce the errors related to shadowing.

Furthermore, the current study remains limited to a selected number of viewing geometries. These results indicate that biases depend on the viewing geometry. Therefore, the methods of the current study should be extended to other geometries to obtain a more general picture of the effect of viewing geometry on retrieval errors arising from three-dimensional cloud-radiation interactions. In terms of PPA, the transmittance biases plotted in Fig. 16 already provide a first indication of the sign and magnitude of this error across a range of viewing geometries.